Pendant Vertices Research Articles

On two eccentricity-based topological indices of graphs

For a connected graph G, the eccentric connectivity index (ECI) and connective eccentricity index (CEI) of G are, respectively, defined as ξc(G)=∑vi∈V(G)degG(vi)εG(vi), ξce(G)=∑vi∈V(G)degG(vi)εG(vi) where degG(vi) is the degree of vi in G and εG(vi) denotes the eccentricity of vertex vi in G. In this paper we study on the difference of ECI and CEI of graphs G, denoted by ξD(G)=ξc(G)−ξce(G). We determine the upper and lower bounds on ξD(T) and the corresponding extremal trees among all trees of order n. Moreover, the extremal trees with respect to ξD are completely characterized among all trees with given diameter d. And we also characterize some extremal general graphs with respect to ξD. Finally we propose that some comparative relations between CEI and ECI are proposed on general graphs with given number of pendant vertices.

Maximizing Wiener index for trees with given vertex weight and degree sequences

The Wiener index is maximized over the set of trees with the given vertex weight and degree sequences. This model covers the traditional “unweighed” Wiener index, the terminal Wiener index, and the vertex distance index. It is shown that there exists an optimal caterpillar. If weights of internal vertices increase in their degrees, then an optimal caterpillar exists with weights of internal vertices on its backbone monotonously increasing from some central point to the ends of the backbone, and the same is true for pendent vertices. A tight upper bound of the Wiener index value is proposed and an efficient greedy heuristics is developed that approximates well the optimal index value. Finally, a branch and bound algorithm is built and tested for the exact solution of this NP-complete problem.

Signed graphs with cut points whose positive inertia indexes are two

A signed graph Gσ consists of an underlying graph G and a sign function σ, which assigns each edge uv of G a sign σ(uv), either positive or negative. The adjacency matrix of Gσ is defined as A(Gσ)=(au,vσ) with au,vσ=σ(uv)au,v, where au,v=1 if u,v∈V(G) are adjacent, and au,v=0 otherwise. The positive inertia index of Gσ, written as p(Gσ), is defined to be the number of positive eigenvalues of A(Gσ). Recently, Yu et al. (2016) [12] characterized the signed graphs Gσ with pendant vertices such that p(Gσ)=2. In this paper, we extend the above work to a more general case, characterizing the signed graphs Gσ with cut points whose positive inertia index is 2.

On the maximum ABC index of graphs without pendent vertices

Let G be a simple graph. The atom–bond connectivity index (ABC) of G is defined as ABC(G)=∑uv∈E(G)d(u)+d(v)−2d(u)d(v), where d(v) denotes the degree of vertex v of G. We characterize the graphs with n vertices, minimum vertex degree ≥ 2, and m edges for m=2n−4 and m=2n−3, that have maximum ABC index.

Convex and quasiconvex functions on trees and their applications

We introduce convex and quasiconvex functions on trees and prove that for a tree the eccentricity, transmission and weight functions are strictly quasiconvex. It is shown that the Perron vector of the distance matrix is strictly convex whereas the Perron vector of the distance signless Laplacian is quasiconvex for a tree. In the class of all trees with a given number of pendent vertices, we prove that the distance Laplacian and distance signless Laplacian spectral radius are both maximized at a dumbbell. Among all trees with fixed maximum degree, we prove that the broom is the unique tree that maximizes the distance Laplacian and distance signless Laplacian spectral radius. We find the unique graph that maximizes the distance spectral radius in the class of all unicyclic graphs of girth g on n vertices. Also we find the unique graph that maximizes the distance signless Laplacian and the distance Laplacian spectral radius in the class of all unicyclic graphs on n vertices.

Spectral radius and extremal graphs for class of unicyclic graph with pendant vertices

In this article, we research on the spectral radius of extremal graphs for the unicyclic graphs with girth g mainly by the graft transformation and matching and obtain the upper bounds of the spectral radius of unicyclic graphs.

Isotropic Matroids III: Connectivity

The isotropic matroid $M[IAS(G)]$ of a graph $G$ is a binary matroid, which is equivalent to the isotropic system introduced by Bouchet. In this paper we discuss four notions of connectivity related to isotropic matroids and isotropic systems. We show that the isotropic system connectivity defined by Bouchet is equivalent to vertical connectivity of $M[IAS(G)]$, and if $G$ has at least four vertices, then $M[IAS(G)]$ is vertically 5-connected if and only if $G$ is prime (in the sense of Cunningham's split decomposition). We also show that $M[IAS(G)]$ is $3$-connected if and only if $G$ is connected and has neither a pendant vertex nor a pair of twin vertices. Our most interesting theorem is that if $G$ has $n\geq7$ vertices then $M[IAS(G)]$ is not vertically $n$-connected. This abstract-seeming result is equivalent to the more concrete assertion that $G$ is locally equivalent to a graph with a vertex of degree $<\frac{n-1}{2}$.

Applications of Estrada indices and energy to a family of compound graphs

To track the gradual change of the adjacency matrix of a simple graph G into the signless Laplacian matrix, V. Nikiforov in [35] suggested the study of the convex linear combination Aα (α-adjacency matrix),Aα(G)=αD(G)+(1−α)A(G), for α∈[0,1], where A(G) and D(G) are the adjacency and the diagonal vertex degrees matrices of G, respectively. Taking this definition as an idea the next matrix was considered for a,b∈R. The matrix Aa,b defined byAa,b(G)=aD(G)+bA(G), extends the previous α-adjacency matrix. This matrix is designated the (a,b)-adjacency matrix ofG. Both adjacency matrices are examples of universal matrices already studied by W. Haemers. In this paper, we study the (a,b)-adjacency spectra for a family of compound graphs formed by disjoint balanced trees whose roots are identified to the vertices of a given graph. In consequence, new families of cospectral (adjacency, Laplacian and signless Laplacian) graphs, new hypoenergetic graphs (graphs whose energy is less than its vertex number) and new explicit formulae for Estrada, signless Laplacian Estrada and Laplacian Estrada indices of graphs were obtained. Moreover, sharp upper bounds of the above indices for caterpillars, in terms of length of the path and of the maximum number of its pendant vertices, are given.

On extremal cacti with respect to the Szeged index

The Szeged index of a graph G is defined as Sz(G)=∑e=uv∈Enu(e)nv(e), where nu(e) and nv(e) are, respectively, the number of vertices of G lying closer to vertex u than to vertex v and the number of vertices of G lying closer to vertex v than to vertex u. A cactus is a graph in which any two cycles have at most one common vertex. Let C(n,k) denote the class of all cacti with order n and k cycles, and Cnt denote the class of all cacti with order n and t pendant vertices. In this paper, a lower bound of the Szeged index for cacti of order n with k cycles is determined, and all the graphs that achieve the lower bound are identified. As well, the unique graph in Cnt with minimum Szeged index is characterized.

Some new graceful generalized classes of diameter six trees

Here we denote a {\it diameter six tree} by $(c; a_{1}, a_{2}, \ldots, a_{m}; b_{1}, b_{2}, \ldots, b_{n}; c_{1}, c_{2}, \ldots, c_{r})$, where $c$ is the center of the tree; $a_{i}, i = 1, 2, \ldots, m$, $b_{j}, j = 1, 2, \ldots, n$, and $c_{k}, k = 1, 2, \ldots, r$ are the vertices of the tree adjacent to $c$; each $a_{i}$ is the center of a diameter four tree, each $b_{j}$ is the center of a star, and each $c_{k}$ is a pendant vertex. Here we give graceful labelings to some new classes of diameter six trees $(c; a_{1}, a_{2}, \ldots, a_{m}; b_{1}, b_{2}, \ldots, b_{n}; c_{1}, c_{2}, \ldots, c_{r})$ in which a diameter four tree may contain any combination of branches with the total number of branches odd though with some conditions on the number of odd, even, and pendant branches. Here by a branch we mean a star, i.e. we call a star an odd branch if its center has an odd degree, an even branch if its center has an even degree, and a pendant branch if it is a pendant vertex.

On the general sum-connectivity index of trees with given number of pendent vertices

The general sum-connectivity index of a graph G is defined as χα(G)=∑uv∈E(G)(d(u)+d(v))α, where d(u) denotes the degree of a vertex u in G and α is a real number. In this paper, we present the maximum general sum-connectivity indices of trees and chemical trees with given number of pendent vertices for −1≤α<0. The corresponding extremal graphs are also characterized.

Some New Graceful Lobsters with Pendant Vertices with Central Paths

The objective of this article is to give graceful labeling to some new classes of lobsters in a bid to resolve the three and half decade old Bermond’s conjecture that all lobsters are graceful. Here we use the method of component moving transformation such as the transfer of the first and second type and the derived transformations such as BD8TF, 1JTF, and 2JTF for generating graceful trees from a given one. In a bid to resolve Bermond’s conjecture here we give graceful labelings to many classes of lobsters possessing at least one of the two distinct features from those found in the literature as detailed below. 1.The central paths of the lobsters contain one or more vertices which do not have any neighbour apart from those on the central path. 2.One or more vertices of the central paths are attached to leaves. 3. The vertices on the central path may be attached to any of the fifteen different combination of odd, even, and pendant branches.

ON CONSTRUCTIONS OF NEW SUPER EDGE-MAGIC GRAPHS FROM SOME OLD ONES BY ATTACHING SOME PENDANTS

Baskoro et al. [1] provided some constructions of new super edge-magic graphs from some old ones by attaching 1, 2, or 3 pendant vertices and edges. In this paper, we introduce (q, m)-super edge-magic total labeling and we give a construction of new super edge-magic graphs by attaching n pendant vertices and edges under some conditions, for any positive integer n. Also, we give a constraint on our construction.

The general Randić index of trees with given number of pendent vertices

The general Randić index of a graph G is defined as Rα(G)=∑uv∈E(G)(d(u)d(v))α, where d(u) denotes the degree of a vertex u in G and α is a real number. In this paper, we determine the maximum general Randić indices of trees and chemical trees with n vertices and k pendent vertices for 4≤k≤⌊n+23⌋ and α0 ≤ α < 0, where α0≈−0.5122 is the unique non-zero root of the equation 6·4α−20·9α+10·12α−16α+5·24α=0. The corresponding extremal graphs are also characterized.

Maximum Laplacian energy of unicyclic graphs

We study the Laplacian and the signless Laplacian energy of connected unicyclic graphs, obtaining a tight upper bound for this class of graphs. We also find the connected unicyclic graph on n vertices having largest (signless) Laplacian energy for 3≤n≤13. For n≥11, we conjecture that the graph consisting of a triangle together with n−3 balanced distributed pendent vertices is the candidate having the maximum (signless) Laplacian energy among connected unicyclic graphs on n vertices. We prove this conjecture for many classes of graphs, depending on σ, the number of (signless) Laplacian eigenvalues bigger than or equal to 2.

Potential eventual positivity of sign patterns with the underlying broom graph

A sign pattern is a matrix whose entries belong to the set {+,−,0}. An n-by-nsign pattern A is said to allow an eventually positive matrix or be potentially eventually positive if there exist at least one real matrix A with the same sign pattern as A and a positive integer k0 such that Ak>0 for all k≥k0. Identifying the necessary and sufficient conditions for an n-by-n sign pattern to be potentially eventually positive, and classifying the n-by-n sign patterns that allow an eventually positive matrix were posed as two open problems by Berman, Catral, Dealba, et al. In this article, we focus on the potential eventual positivity of a collection of the n-by-n tree sign patterns An,4 whose underlying graph G(An,4) consists of a path P with 4 vertices, together with (n−4) pendent vertices all adjacent to the same end vertex of P. Some necessary conditions for the n-by-n tree sign patterns An,4 to be potentially eventually positive are established. All the minimal subpatterns of An,4 that allow an eventually positive matrix are identified. Consequently, all the potentially eventually positive subpatterns of An,4 are classified.

A class of diameter six trees exhibiting graceful labeling

A diameter six tree may be represented by a set of vertices, say (a0; a1, a2, . , am; b1, b2, . , bn; c1, c2, . , cr), where a0 stands for the center of the tree and the vertices ai, i = 1, 2, . , m, bj, j = 1, 2, . , n, ck, k = 1, 2, . , r are adjacent to a0. Moreover, each ai is the center of a diameter four tree, each bj is the center of a star, and each ck is a pendant vertex. Observe that in a diameter four tree, the neighbors of the center vertex are either the centers of some stars or pendant vertices. Here we call a star an odd branch if its center has an odd degree and an even branch if its center has an even degree. In the diameter six trees (a0; a1, a2, . , am; b1, b2, . , bn; c1, c2, . , cr) to which we give graceful labeling in this paper, each vertex ai, 1 ≤ i ≤ m, has an even degree and it may be adjacent to four different combinations consisting of odd branches, even branches, and pendant vertices.

Further results regarding the sum of domination number and average eccentricity

The average eccentricity of a graph G, denoted by ecc(G), is the mean value of eccentricities of all vertices of G. Let Dn, i be the n-vertex tree obtained from a path Pn−1=v1v2⋯vn−1 by attaching a pendent vertex to vi. In [13], it was shown that the maximum value for the sum of domination number and average eccentricity among n-vertex (connected) graphs is attained by Dn, 3 when n≡0(mod3), and attained by the path Pn when n¬≡0(mod3). In this paper, we will further determine the second maximum value for the sum of domination number and average eccentricity among n-vertex (connected) graphs. It is interesting that the graphs attaining that second maximum value have three cases, which is Dn, 6 when n≡0(mod3),Dn, 3 when n≡1(mod3), and Tn when n≡2(mod3), where Tn is the n-vertex tree obtained from a path Pn−2=v1v2⋯vn−2 by attaching a pendent vertex to v3, and a pendent vertex to vn−4.

Strong reciprocal eigenvalue property of a class of weighted graphs

Let H be the class of connected bipartite graphs G with a unique perfect matching M. For G∈H, let WG be the set of weight functions w on the edge set E(G) such that w(e)=1 for each matching edge and w(e)>0 for each nonmatching edge. Let Gw denote the weighted graph with G∈H and w∈WG. The graph Gw is said to satisfy the reciprocal eigenvalue property, property (R), if 1/λ is an eigenvalue of the adjacency matrix A(Gw) whenever λ is an eigenvalue of A(Gw). Moreover, if the multiplicities of the reciprocal eigenvalues are the same, we say Gw has the strong reciprocal eigenvalue property, property (SR). Let Hg={G∈H|G/M is bipartite}, where G/M is the graph obtained from G by contracting each edge in M to a vertex.Recently in [12], it was shown that if G∈Hg, then Gw has property (SR) for some w∈WG if and only if Gw has property (SR) for each w∈WG if and only if G is a corona graph (obtained from another graph H by adding a new pendant vertex to each vertex of H).Now we have the following questions. Is there a graph G∈H∖Hg such that Gw has property (SR) for each w∈WG? Are there graphs G∈H∖Hg such that Gw never has property (SR), not even for one w∈WG? Are there graphs G∈H such that Gw has property (SR) for some w∈WG but not for all w∈WG? In this article, we supply answers to these three questions. We also supply a graph class larger than Hg where for any graph G, if Gw has property (SR) for one w∈WG, then G is a corona graph.

Sharp bounds on the inverse sum indeg index

The inverse sum indeg index is a recently-introduced bond-additive descriptor that was selected by Vukičević and Gašperov (2010) as a significant predictor of total surface area of octane isomers. In this paper, we present several sharp upper and lower bounds on the inverse sum indeg index in terms of some graph parameters such as the order, size, radius, number of pendant vertices, minimal and maximal vertex degrees, and minimal non-pendent vertex degree, and relate this index to various well-known graph invariants such as the ordinary and multiplicative Zagreb indices, Randić indices, sum-connectivity index, modified Zagreb index, harmonic index, forgotten index, and eccentric connectivity index.